(Thanks for the ping, @Pierre-Yves Gaillard...)
As in Pierre-Yves Gaillard's answer, I think that the usual formalization of "Yoneda's Lemma" does not literally yield the exactness assertion quoted. However, if one tolerates a certain amount of figurative reference, it's not completely ridiculous to refer to using $X\rightarrow Hom(X,-)$, and inferences about the original category from the image, as Yoneda-ish. After all, isn't that why we care about the Yoneda maps?
As Pierre-Yves G notes, and as in @Agusti Roig's elaboration of the key point of the argument, there are further ingredients in play, such as additivity or abelian-ness, which were built into the example scenario my note treated, so I did not have to name them. Thus, the pure Yoneda Lemma is not directly addressing all the issues. However, the idea to "consider" $X\rightarrow Hom(X,-)$ is the watershed, I think. If we take the simplest illustrative situation, abelian groups, so that ("by luck") the Hom(,)'s are in the same category, etc., one can immediately proceed to "do the thing" with no formalism whatsoever. One of the nice didactic points is that we really, really do need the "naturality" to know that the squares commute, to reach the conclusion. This is the simplest illustration I know (apart from the usual algebraic topology examples) of the on-the-street genuine content of "naturality", as opposed to the all-too-common cocktail-party misunderstanding of it as "not making arbitrary choices", etc.
But, yes, then the narrative was not connected to any formalism, either. The features of $X\rightarrow Hom(X,-)$ that made things work were not abstracted or formalized.
I'd plead guilty to "formal incorrectness", but with extenuating circumstances, as follows. Namely, the point of the linked-to note was to illustrate the immediate utility of $X\rightarrow Hom(X,-)$ to very tangible issues, such as showing right-exactness of $-\otimes X$, by seeing that it is a left adjoint, and showing left adjoints (with additivity...) are right exact, for very general reasons. The intended audience had/has little prior acquaintance with category theory or homological algebra, and that note was intended as an advertisement or promotion of such ideas, without formalizing those ideas enough to name them precisely. If I recall correctly, in particular, someone had been wrangling with a direct argument for the right exactness of tensor-product, and had gotten bogged-down in details that shouldn't matter, and I was trying to make the case to them that the causality in the situation could (without paying too heavy a price) be better understood as being about adjoint functors, etc. To make that case, there was absolutely no room for any set-up. At the same time, the name-dropping seemed appropriate on philosophical grounds.
I am in no sense "a category theorist" or "homological algebraist", but do have some appreciation for the widespread usefulness of the ideas, "even" without formalism or abstraction of categorical notions. Indeed, I would claim that the ideas "are there", whether one recognizes/formalizes them or not, and that recognizing them is immediately useful in the most pragmatic sense. Here I am playing against a common belief that one "chooses", or not, to "do" category theory or homological algebra, and that ignoring those ideas is a viable, completely sensible option.
In that context, perhaps over-selling "Yoneda" by throwing in a few further ingredients is an excess that can be forgiven? (Not quite as bad as a "stone soup" scenario.)
I do also admit to an ever-waning interest in formalization, especially definitions. Pithy examples seem to me much better, even without a pre-existing name for the phenomenon illustrated. One "problem" with this is that naturally-occurring illustrative examples often are not "pure", in the sense that more than the single phenomenon is involved. E.g., if I say I'm illustrating Yoneda by the exactness argument at hand, the illustration is not "pure", and there is potential confusion, indeed. On the other hand, if I'm trying to convince someone of the virtues of an idea (the $X\rightarrow Hom(X,-)$), it would be unwise to deliberately ignore the "extra" features they actually cared about.
In summary: indeed, Yoneda does not literally entail that exactness conclusion, I think. Nor need it be invoked in the simple case at hand. Rather some aspects of a special case of Yoneda are being proved directly, along with some additional bits to address the problem at hand.
Edit: in response to @Bruno Stonek's follow-up question, the "naturality" is the obvious, essentially trivial assertion that exactness of the top row entails exactness of the bottom row in the following:
$$
\matrix{
0 & \rightarrow & Hom(LX,A) & \rightarrow & Hom(LX,B) & \rightarrow & Hom(LX,C)
\cr
& & \downarrow & & \downarrow & & \downarrow
\cr
0 & \rightarrow & Hom(X,RA) & \rightarrow & Hom(X,RB) & \rightarrow & Hom(X,RC)
}
$$
where the vertical arrows are isomorphisms, and $L,R$ are left and right adjoint to each other. That is, without naturality (or whatever name one likes), the vertical isomorphisms would not necessarily assure that the squares commute.
Another Edit: in response to Bruno Stonek's further "tangential" question... The "additivity" or some similar qualifier is necessary for "exactness" to make sense at all. In familiar concrete categories where we'd have any impulse to mention it, such as categories of modules over some ring, it's just structure we're used-to. Not every category behaves this way, of course. Sufficient notions to talk about kernels and cokernels can obviously be axiomatized...
Best Answer
The $\text{Hom}$ functor preserves limits in each argument (in a very strong sense), neither preserves colimits in general. You should prove this as it's the source of continuity for most other things that are continuous, most notably adjoints. (Because (co)limits in functor categories are computed point-wise, this lifts to the Yoneda embeddings.)
But be a bit careful, $\text{Hom}(-,X)$ is a functor $\mathcal{C}^{op} \to \mathbf{Set}$, so being continuous means it takes colimits in $\mathcal{C}$ to limits in $\mathbf{Set}$. (Of course, this is equivalent to taking limits in $\mathcal{C}^{op}$ to limits in $\mathbf{Set}$, i.e. continuity.)
Considering a slightly different problem, if we fix an $X$ and ask what does it mean for $\text{Hom}(X,-)$ to preserve all colimits, you get the notion of a tiny object assuming $\mathcal{C}$ has all colimits (this situation is the same as the one you were talking about, just in the opposite category). An object being tiny is an unusual thing, for example, in $\mathbf{Set}$, the only tiny object is the terminal object. It would be unusual for a cocomplete category to have nothing but tiny objects. In fact, considering different classes of colimits gives different notions of "smallness". For example, if $\text{Hom}(X,-)$ preserves all filtered colimits, then $X$ is a compact or finitely presented object. A strongly finitely presented object is one for which that functor preserves all sifted colimits. A connected object (in an extensive category) is one where that functor preserves coproducts. A duality relative to a limit doctrine gives some general definitions and results for situations like this.